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Theorem eqelssd 3120
Description: Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)
Hypotheses
Ref Expression
eqelssd.1  |-  ( ph  ->  A  C_  B )
eqelssd.2  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
Assertion
Ref Expression
eqelssd  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, A    x, B    ph, x

Proof of Theorem eqelssd
StepHypRef Expression
1 eqelssd.1 . 2  |-  ( ph  ->  A  C_  B )
2 eqelssd.2 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
32ex 114 . . 3  |-  ( ph  ->  ( x  e.  B  ->  x  e.  A ) )
43ssrdv 3107 . 2  |-  ( ph  ->  B  C_  A )
51, 4eqssd 3118 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481    C_ wss 3075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3081  df-ss 3088
This theorem is referenced by:  fiuni  6873  ennnfonelemrn  11966  ennnfonelemdm  11967  unirnblps  12628  unirnbl  12629  dvidlemap  12866  dviaddf  12875  dvimulf  12876  dvcj  12879  dvrecap  12883
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